Is there any way to change the mouse icon with having the user select a tool/hopperbin?

Current, PlayerMouse.Icon doesn't work, so we have to set it via the mouse derived from a Tool or HopperBin event. I've heard that equipping the tool using EquipTool and then removing it won't keep the icon, and you can't force select a HopperBin (as far as my knowledge).

Is there any way to do that without having the user having to press a key or using a tool?

At the moment, there is no way, but what you can do is have the HopperBin change the mouse Icon when selected, and then remove itself, so the Icon will stay like that until the player resets.